# [EM] Some Voting Tables

Forest Simmons fsimmons at pcc.edu
Thu Dec 13 10:27:58 PST 2001

```Here's an example that turns out to be more interesting than it first
appears to be:

(Sincere intensities or utilities are in parentheses.)

45  A(100) B(50) C(0)
30  B(100) C(50) A(0)
25  C(100) A(50) B(0)

If the voters in the three factions all vote sincerely (lacking
information on which to base sophisticated strategy), then candidate A
wins under the following methods:

Lone mark plurality (the most commonly accepted method), Single
Transferable Vote (STV a.k.a. IRV), Borda Count (i.e. highest average
rank), Cardinal Ratings (i.e. highest average rating), Coombs (sequential
elimination of candidates with most last place votes [absent clear
majority winner]), Ranked Pairs (locking in the two highest margin
defeat), etc.

In all of these methods candidate C comes in last place.

[Is there any well known method besides Bucklin that would not give the
group ranking as  A > B > C ? ]

In this "zero information" case Approval voters have to decide whether to
use above mean inclusive strategy or above mean exclusive strategy. i.e.
should they approve their mean candidates or not?

If approximately half of each faction goes each way, then candidate A will
also be the Approval winner. This could come about through coin tossing or
by pro-active collusion within each faction.

However, that much potential for vote coordination would definitely take
us out of the zero information case.

So let us now consider the other (more interesting) extreme, the perfect
information case.

Who would be the various winners in each of the various methods in the
perfect information case (assuming sophisticated voters)?

It seems to me, for example, that under Borda or IRV/STV, the B faction
voters would see the handwriting on the wall and be tempted to vote C
above B to keep A from winning.

Although I haven't done a precise game theoretic analysis on this yet, it
seems to me that B has the power and incentive to keep A from winning, and
that C would know this and not mess things up by supporting A.

With perfect information it seems likely that under IRV and Borda the
voters would end up picking the consensus sincere last choice C.

In the perfect information case how would the various candidates fair
under Plurality, Ranked Pairs, Coombs, etc.?

Under Approval or Cardinal Ratings candidate B would be the likely winner.
If enough interest is shown, I will give the logic behind this conclusion
in a later posting.

I will finish this posting by giving a scenario in which an example like
this could arise:

Candidates A, B, and C have the tabulated (and openly acknowledged)
sincere ratings for each other, and have been designated as proxies for
45%, 30%, and 25% of the voters, respectively, in order to decide the
winner.

Forest

```